On Some Characterization Theorems for New Classes of Multiple-Objective Control Models

Abstract

This paper introduces a new class of multiple-objective control models driven by path-independent curvilinear integrals involving the partial derivatives of the control variable. We investigate its solution set by considering a dual problem. Various duality results are formulated and proved in order to study and investigate the relationships between the set of solutions for these two variational control problems. Specifically, first, we establish that the value of the cost functional associated with the primal model cannot be greater than the value of the cost functional associated with the dual model. Secondly, the following two results present a strong-type duality between the variational models considered. At the end, we illustrate the main findings of the current paper with a numerical example.

1. Introduction

We all know the importance and efficiency of duality theory in the study of extremization problems. Mishra and Mukherjee [1] used the Pareto optimum concept to analyze a duality theory for classical multiobjective variational problems. Wolfe dual and Mond–Weir dual have been considered, under the generalized $(F,\rho )$-convexity hypotheses, for establishing weak and strong duality theorems. Later, Bhatia and Mehra [2] established sufficient optimality conditions and duality theorems for multiobjective continuous-time variational problems under B-type I and generalized B-type I assumptions of the involved functions. Also, by using invexity assumptions, Nahak and Nanda [3] formulated Wolfe and Mond–Wei duals for a family of non-differentiable multiple-objective variational models. Xiuhong [4], taking into account the innovative ideas formulated in Leitmann [5] and Swam [6], stated a duality for some multiobjective control problems. Pereira [7,8] investigated the control design associated with autonomous vehicles and provided a maximum type principle for state-constrained impulsive control problems. In 2000, a generalized invexity was proposed for the duality theory associated with multi-objective programming problems (see Aghezzaf Hachimi [9]). Efficiency and duality outcomes for some classes of multiobjective control problems with generalized invexity have been stated by Reddy and Mukherjee [10] and Zhian et al. [11]. Moreover, Kim and Kim [12] formulated generalized type I invexity to investigate duality in vector optimization. Antczak [13,14] studied G-invex variational problems by providing both (necessary and sufficient) optimality criteria and the associated duality theory. In order to state a necessary and sufficient condition such that each Kuhn–Tucker point is an optimal solution, Arana-Jiménez et al. [15,16] studied KT-invex and FJ-invex control-type problems. Hachimi and Aghezzaf [17] established sufficiency and duality in multiobjective optimization problems governed by generalized type I functions. Later, Khazafi et al. [18,19] extended the previous point of view and introduced a generalized type I univexity and $(B,\rho )$-type I functions. Zhang et al. [20] established sufficiency criteria and duality results for multi-objective control problems with G-invexity assumptions. Recently, Treanţă [21] formulated well-posedness results in some optimization problems involving variational inequality constraints. Moreover, Saeed et al. [22] stated optimality conditions for multiobjective minimization models implying semi-infinite constraints and generalized $(h,\phi )-G$-type I functions. Abdulaleem and Treanţă [23] formulated and proved optimality criteria and a duality in E-differentiable multiobjective problems with $V-E$-type I functions. Very recently, Marghescu and Treanţă [24] studied a family of control problems generated by approximately pseudo-convex multiple integral functionals.

In this paper, we continue and extend the analysis performed in Treanţă et al. [25], where the authors stated and proved necessary and sufficient criteria of efficiency for a feasible point in a class of controlled variational models involving multiple integral-type cost functionals. Concretely, based on a class of multi-objective variational control models involving path-independent curvilinear integrals (not multiple integrals as in Treanţă et al. [25]), we introduce a dual problem. In addition, various duality results are formulated and proved in order to study and investigate the relationships between the set of solutions for these two variational control problems. The principal novelty elements included in the present paper are given by the original and innovative ideas and the new mathematical framework used to prove the main theorems (the presence of partial derivatives of the control variable is a new element in this research area). Also, due to the applications in physics of the functionals used in this study (namely, second-type curvilinear integrals calculate the mechanical work conducted by a variable force to move its point of application along the considered curve), the present work has various implications in control theory, robotics, economic systems, or energy optimization. Furthermore, the path independence of the considered integrals implies a symmetry of the partial derivatives associated with the functionals in question.

2. Problem Formulation

In the following, we state notation and definitions that will be used to establish the principal results associated with this paper. In this regard, for any two vectors, $\zeta ={\left({\zeta }_1,{\zeta }_2,\dots ,{\zeta }_w\right)}^T,\mu ={\left({\mu }_1,{\mu }_2,\dots ,{\mu }_w\right)}^T$, we consider the following rules:

(i) $\zeta =\mu$ if and only if ${\zeta }_r={\mu }_r$ for all $r=1,\dots ,w$;

(ii) $\zeta <\mu$ if and only if ${\zeta }_r<{\mu }_r$ for all $r=1,\dots ,w$;

(iii) $\zeta \leqq \mu$ if and only if ${\zeta }_r\le {\mu }_r$ for all $r=1,\dots ,w$;

(iv) $\zeta \le \mu$ if and only if $\zeta \leqq \mu$ and $\zeta \ne \mu$.

Let $D=D_{a,b}$ be a compact set in ${\mathbb{R}}^j$, $C\subset D$ a piecewise smooth curve that links the points a and b in $D_{a,b}$, and let $U=\{1,\dots ,v\},V=\{1,\dots ,u\}$ be some index sets. Suppose $\tau \left(l\right)$ is a differentiable mapping (piecewise) of $l\in D$, and ${\tau }_{\alpha }\left(l\right):={\displaystyle \frac{\partial \tau }{\partial l^{\alpha }}}\left(l\right)$ is the partial derivative of $\tau \left(l\right)$ related to $l^{\alpha }$ in D. Also, consider $ϵ\left(l\right)$ is a differentiable mapping (piecewise) of $l\in D$, and ${ϵ}_{\beta }\left(l\right):={\displaystyle \frac{\partial ϵ}{\partial l^{\beta }}}\left(l\right)$ is the partial derivative of $ϵ\left(l\right)$ related to $l^{\beta }$ in D. Denote by A the space of state functions $\tau :D\to {\mathbb{R}}^w$ with norm $\parallel \tau \parallel =$${\parallel \tau \parallel }_{\infty }+{\parallel {\tau }_{\alpha }\parallel }_{\infty }$, and denote by Y the space of control functions $ϵ:D\to {\mathbb{R}}^x$ with the uniform norm, as well.

In the following, motivated by the physical applications (mechanical work) of the involved functionals, we study the multi-objective variational control model involving path-independent curvilinear integrals, defined as followed:

$$\begin{array}{cc}\hfill & \left(\mathrm{Primal}\right)\underset{(\tau ,ϵ)}{min}{\int }_C{\lambda }_p(l,\tau \left(l\right),{\tau }_{\alpha }\left(l\right),ϵ\left(l\right),{ϵ}_{\beta }\left(l\right))d{l}^p\hfill \\ \hfill & =\left({\int }_C{\lambda }_p^1(l,\tau \left(l\right),{\tau }_{\alpha }\left(l\right),ϵ\left(l\right),{ϵ}_{\beta }\left(l\right))d{l}^p,\dots ,{\int }_C{\lambda }_p^v(l,\tau \left(l\right),{\tau }_{\alpha }\left(l\right),ϵ\left(l\right),{ϵ}_{\beta }\left(l\right))d{l}^p\right)\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill \\ \hfill & \delta (l,\tau \left(l\right),{\tau }_{\alpha }\left(l\right),ϵ\left(l\right),{ϵ}_{\beta }\left(l\right)):={\tau }_{\alpha }\left(l\right)-h(l,\tau \left(l\right),ϵ\left(l\right),{ϵ}_{\beta }\left(l\right))\leqq 0,l\in D,\hfill \\ \hfill & \tau \left(a\right)=\varphi =\mathrm{given},\tau \left(b\right)=\psi =\mathrm{given},\hfill \end{array}$$

where ${\lambda }_p=\left({\lambda }_p^1,\dots ,{\lambda }_p^v\right):D\times {\mathbb{R}}^w\times {\mathbb{R}}^{wj}\times {\mathbb{R}}^x\times {\mathbb{R}}^{xj}\to {\mathbb{R}}^v,p=\overline{1,j}$, is a continuously differentiable v-dimensional functional, and $\delta :D\times {\mathbb{R}}^w \times$ ${\mathbb{R}}^{wj} \times {\mathbb{R}}^x \times {\mathbb{R}}^{xj}\to {\mathbb{R}}^u$ is a $C^1$-class functional.

Next, we denote $\tau \left(l\right)$ and ${\tau }_{\alpha }\left(l\right)$ as $\tau$ and ${\tau }_{\alpha }$, respectively, and $ϵ\left(l\right)$ and ${ϵ}_{\beta }\left(l\right)$ as $ϵ$ and ${ϵ}_{\beta }$, respectively. Also, we consider the notations

$$\begin{array}{c}{\gamma }_{\tau ϵ}\left(l\right):=(l,\tau \left(l\right),{\tau }_{\alpha }\left(l\right),ϵ\left(l\right),{ϵ}_{\beta }\left(l\right)),{\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right):=(l,\overline{\tau }\left(l\right),{\overline{\tau }}_{\alpha }\left(l\right),\overline{ϵ}\left(l\right),{\overline{ϵ}}_{\beta }\left(l\right)),\\ {\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right):=(l,\tau \left(l\right),\overline{\tau }\left(l\right),ϵ\left(l\right),\overline{ϵ}\left(l\right)).\end{array}$$

Let $\mathcal{K}$ denote the set of all feasible points of (Primal), that is

$$\mathcal{K}=\left\{\right(\tau ,ϵ):\tau \in A,ϵ\in Y\mathrm{satisfying}\mathrm{the}\mathrm{constraints}\mathrm{in}\left(\mathrm{Primal}\right),\mathrm{for}\mathrm{all}l\in D\}.$$

The following three definitions present different types of solutions for the above-mentioned class of extremization problems, considered in the rest of the paper.

Definition 1. 

A pair $(\overline{\tau },\overline{ϵ})\in \mathcal{K}$ is named weakly efficient point of (Primal) if there is no other feasible pair $(\tau ,ϵ)\in \mathcal{K}$ satisfying

$${\int }_C{\lambda }_p\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p<{\int }_C{\lambda }_p\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p.$$

Definition 2. 

A pair $(\overline{\tau },\overline{ϵ})\in \mathcal{K}$ is named efficient point of (Primal) if there is no other feasible pair $(\tau ,ϵ)\in \mathcal{K}$ satisfying

$${\int }_C{\lambda }_p\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\le {\int }_C{\lambda }_p\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p.$$

Definition 3. 

A pair $(\overline{\tau },\overline{ϵ})\in \mathcal{K}$ is named properly efficient point of (Primal) if there is $L>0$ such that the following inequality

$${\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p-{\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p$$

$$\le L\left({\int }_C{\lambda }_p^k\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p-{\int }_C{\lambda }_p^k\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)$$

holds, for each $r\in U$, and for some k, satisfying

$${\displaystyle {\int }_C{\lambda }_p^k\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p>{\int }_C{\lambda }_p^k\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p,}$$

whenever $(\tau ,ϵ)\in \mathcal{K}$ and ${\displaystyle {\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p<{\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p}$.

Next, we recall the classical definition of a strictly increasing function.

Definition 4. 

A real-valued function $\mathsf{\Psi }:\mathbb{R}\to \mathbb{R}$ is named strictly increasing if and only if

$$\forall \zeta ,\mu \in \mathbb{R},\zeta <\mu ⟹\mathsf{\Psi }\left(\zeta \right)<\mathsf{\Psi }\left(\mu \right).$$

The definitions presented below play an essential role in the formulation and demonstration of the basic results obtained in this paper. Specifically, we introduce (for the first time) the concepts of H-type I, strictly H-type I, pseudo-quasi-H-type I, striclty-pseudo-quasi-H-type I, weak-pseudo-quasi-H-type I, strong-pseudo-quasi-H-type I, and weak-striclty-pseudo-quasi-H-type I for the pair $({\lambda }_p,\delta )$, where ${\lambda }_p,\delta$ are defined as above.

Let $Im\left(F^r\right),r\in U$, be the range of

$${\displaystyle F^r:A\times Y\to \mathbb{R},F^r(\tau ,ϵ):={\int }_C{\lambda }_p^r(l,\tau \left(l\right),{\tau }_{\alpha }\left(l\right),ϵ\left(l\right),{ϵ}_{\beta }\left(l\right))d{l}^p,}$$

where $\tau \in A,ϵ\in Y$, and $Im\left(G^q\right),q\in V$, be the range of

$${\displaystyle G^q:A\times Y\to \mathbb{R},G^q(\tau ,ϵ):={\int }_C{\delta }^q(l,\tau \left(l\right),{\tau }_{\alpha }\left(l\right),ϵ\left(l\right),{ϵ}_{\beta }\left(l\right))d{l}^p,}$$

where $\tau \in A,ϵ\in Y$.

Definition 5. 

For $(\overline{\tau },\overline{ϵ})\in A\times Y$, if there exists a smooth vector-valued functional $H_{{\lambda }_p}=\left(H_{{\lambda }_p^1},\dots ,H_{{\lambda }_p^v}\right):\mathbb{R}\to {\mathbb{R}}^v$ with $H_{{\lambda }_p^r}:Im\left(F^r\right)\to \mathbb{R}$ as a strictly increasing function, a smooth vector-valued functional $H_{\delta }=\left(H_{{\delta }^1},\dots ,H_{{\delta }^u}\right):\mathbb{R}\to {\mathbb{R}}^u$ with $H_{{\delta }^q}:Im\left(G^q\right)\to \mathbb{R}$ as a strictly increasing function, $\nu :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^w$ with $\nu (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\nu |}_{l=a,b}=0$, and $\vartheta :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^x$ with $\vartheta (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\vartheta |}_{l=a,b}=0$, such that

$$\begin{array}{cc}\hfill & H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)-H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & \ge H_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right\}d{l}^p,r\in U,\hfill \end{array}$$

(1)

and

$$\begin{array}{cc}\hfill & -H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & \ge H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{\tau }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{{\tau }_{\alpha }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{ϵ}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{{ϵ}_{\beta }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right\}d{l}^p,q\in V,\hfill \end{array}$$

(2)

hold, for all $(\tau ,ϵ)\in A\times Y$, then the pair $({\lambda }_p,\delta )$ is named H-type I at $(\overline{\tau },\overline{ϵ})\in A\times Y$ on $A\times Y$ (related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ).

If (1) and (2) are fulfilled for each $(\overline{\tau },\overline{ϵ})\in A\times Y$, then the pair $({\lambda }_p,\delta )$ is named H-type I on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta }$, ν and ϑ.

Definition 6. 

For $(\overline{\tau },\overline{ϵ})\in A\times Y$, if there exists a smooth vector-valued functional $H_{{\lambda }_p}=\left(H_{{\lambda }_p^1},\dots ,H_{{\lambda }_p^v}\right):\mathbb{R}\to {\mathbb{R}}^v$ with $H_{{\lambda }_p^r}:Im\left(F^r\right)\to \mathbb{R}$ as a strictly increasing function, a smooth vector-valued functional $H_{\delta }=\left(H_{{\delta }^1},\dots ,H_{{\delta }^u}\right):\mathbb{R}\to {\mathbb{R}}^u$ with $H_{{\delta }^q}:Im\left(G^q\right)\to \mathbb{R}$ as a strictly increasing function, $\nu :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^w$ with $\nu (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\nu |}_{l=a,b}=0$, and $\vartheta :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^x$ with $\vartheta (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\vartheta |}_{l=a,b}=0$, such that the following inequalities

$$\begin{array}{cc}\hfill & H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)-H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & >H_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right\}d{l}^p,r\in U,\hfill \end{array}$$

(3)

and

$$\begin{array}{cc}\hfill & -H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & \ge H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{\tau }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{{\tau }_{\alpha }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{ϵ}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{{ϵ}_{\beta }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right\}d{l}^p,q\in V,\hfill \end{array}$$

(4)

hold, for all $(\tau ,ϵ)\in A\times Y,(\tau ,ϵ)\ne (\overline{\tau },\overline{ϵ})$, then the pair $({\lambda }_p,\delta )$ is named strictly-H-type I at $(\overline{\tau },\overline{ϵ})\in A\times Y$ on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

If (3) and (4) are fulfilled for each $(\overline{\tau },\overline{ϵ})\in A\times Y$, then the pair $({\lambda }_p,\delta )$ is named strictly-H-type I on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

Definition 7. 

For $(\overline{\tau },\overline{ϵ})\in A\times Y$, if there exists a smooth vector-valued functional $H_{{\lambda }_p}=\left(H_{{\lambda }_p^1},\dots ,H_{{\lambda }_p^v}\right):\mathbb{R}\to {\mathbb{R}}^v$ with $H_{{\lambda }_p^r}:Im\left(F^r\right)\to \mathbb{R}$ as a strictly increasing function, a smooth vector-valued functional $H_{\delta }=\left(H_{{\delta }^1},\dots ,H_{{\delta }^u}\right):\mathbb{R}\to {\mathbb{R}}^u$ with $H_{{\delta }^q}:Im\left(G^q\right)\to \mathbb{R}$ as a strictly increasing function, $\nu :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^w$ with $\nu (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\nu |}_{l=a,b}=0$, and $\vartheta :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^x$ with $\vartheta (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\vartheta |}_{l=a,b}=0$, such that

$$\begin{array}{cc}\hfill & H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)<H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & ⟹H_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right\}d{l}^p<0,r\in U,\hfill \end{array}$$

(5)

and

$$\begin{array}{cc}\hfill & -H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\le 0\hfill \\ \hfill & ⟹H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{\tau }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{{\tau }_{\alpha }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{ϵ}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{{ϵ}_{\beta }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right\}d{l}^p\le 0,q\in V,\hfill \end{array}$$

(6)

hold, for all $(\tau ,ϵ)\in A\times Y$, then $({\lambda }_p,\delta )$ is named pseudo-quasi-H-type I at $(\overline{\tau },\overline{ϵ})\in A\times Y$ on $A\times Y$ (related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ).

If (5) and (6) are fulfilled for each $(\overline{\tau },\overline{ϵ})\in A\times Y$, then the pair $({\lambda }_p,\delta )$ is named pseudo-quasi-H-type I on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

Definition 8. 

For $(\overline{\tau },\overline{ϵ})\in A\times Y$, if there exists a smooth vector-valued functional $H_{{\lambda }_p}=\left(H_{{\lambda }_p^1},\dots ,H_{{\lambda }_p^v}\right):\mathbb{R}\to {\mathbb{R}}^v$ with $H_{{\lambda }_p^r}:Im\left(F^r\right)\to \mathbb{R}$ as a strictly increasing function, a smooth vector-valued functional $H_{\delta }=\left(H_{{\delta }^1},\dots ,H_{{\delta }^u}\right):\mathbb{R}\to {\mathbb{R}}^u$ with $H_{{\delta }^q}:Im\left(G^q\right)\to \mathbb{R}$ as a strictly increasing function, $\nu :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^w$ with $\nu (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\nu |}_{l=a,b}=0$, and $\vartheta :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^x$ with $\vartheta (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\vartheta |}_{l=a,b}=0$, such that

$$\begin{array}{cc}\hfill & H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)\le H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & ⟹H_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right\}d{l}^p<0,r\in U,\hfill \end{array}$$

(7)

and

$$\begin{array}{cc}\hfill & -H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\le 0\hfill \\ \hfill & ⟹H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{\tau }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{{\tau }_{\alpha }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{{ϵ}_{\beta }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right\}d{l}^p\le 0,q\in V,\hfill \end{array}$$

(8)

hold, for all $(\tau ,ϵ)\in A\times Y,(\tau ,ϵ)\ne (\overline{\tau },\overline{ϵ})$, then $({\lambda }_p,\delta )$ is named strictly-pseudo-quasi-H-type I at $(\overline{\tau },\overline{ϵ})\in A\times Y$ on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

If (7) and (8) are fulfilled for each $(\overline{\tau },\overline{ϵ})\in A\times Y$, then the pair $({\lambda }_p,\delta )$ is named strictly-pseudo-quasi-H-type I on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

Definition 9. 

For $(\overline{\tau },\overline{ϵ})\in A\times Y$, if there exists a smooth vector-valued functional $H_{{\lambda }_p}=\left(H_{{\lambda }_p^1},\dots ,H_{{\lambda }_p^v}\right):\mathbb{R}\to {\mathbb{R}}^v$ with $H_{{\lambda }_p^r}:Im\left(F^r\right)\to \mathbb{R}$ as a strictly increasing function, a smooth vector-valued functional $H_{\delta }=\left(H_{{\delta }^1},\dots ,H_{{\delta }^u}\right):\mathbb{R}\to {\mathbb{R}}^u$ with $H_{{\delta }^q}:Im\left(G^q\right)\to \mathbb{R}$ as a strictly increasing function, $\nu :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^w$ with $\nu (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\nu |}_{l=a,b}=0$, and $\vartheta :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^x$ with $\vartheta (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\vartheta |}_{l=a,b}=0$, such that

$$\begin{array}{cc}& H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)<H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\hfill \\ & ⟹\left[\underset{r\in U}{\forall }{H}_{{\lambda }_p^r}^{\prime }\left({\int }_a^b{\lambda }_p^r\left({\gamma }_{\overline{\tau ϵ}}\left(l\right)\right)d{l}^p\right){\int }_a^b\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right\}d{l}^p\le 0\hfill \\ \hfill & \wedge \underset{r\in U}{\exists }{H}_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.\left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right\}d{l}^p<0\right]\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill & -H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\le 0\hfill \\ \hfill & ⟹H_{\delta q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{\tau }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{{\tau }_{\alpha }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{ϵ}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{{ϵ}_{\beta }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right\}d{l}^p\le 0,q\in V,\hfill \end{array}$$

(10)

hold, for all $(\tau ,ϵ)\in A\times Y,(\tau ,ϵ)\ne (\overline{\tau },\overline{ϵ})$, then $({\lambda }_p,\delta )$ is named weak-pseudo-quasi-H-type I at $(\overline{\tau },\overline{ϵ})\in A\times Y$ on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

If (9) and (10) are fulfilled for each $(\overline{\tau },\overline{ϵ})\in A\times Y$, then the pair $({\lambda }_p,\delta )$ is named weak-pseudo-quasi-H-type I on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

Definition 10. 

For $(\overline{\tau },\overline{ϵ})\in A\times Y$, if there exists a smooth vector-valued functional $H_{{\lambda }_p}=\left(H_{{\lambda }_p^1},\dots ,H_{{\lambda }_p^v}\right):\mathbb{R}\to {\mathbb{R}}^v$ with $H_{{\lambda }_p^r}:Im\left(F^r\right)\to \mathbb{R}$ as a strictly increasing function, a smooth vector-valued functional $H_{\delta }=\left(H_{{\delta }^1},\dots ,H_{{\delta }^u}\right):\mathbb{R}\to {\mathbb{R}}^u$ with $H_{{\delta }^q}:Im\left(G^q\right)\to \mathbb{R}$ as a strictly increasing function, $\nu :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^w$ with $\nu (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\nu |}_{l=a,b}=0$, and $\vartheta :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^x$ with $\vartheta (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\vartheta |}_{l=a,b}=0$, such that

$$\begin{array}{cc}& \left[\underset{r\in U}{\forall }{H}_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)\le H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\right.\hfill \\ & \left.\wedge \underset{r\in U}{\exists }{H}_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)<H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\right]\hfill \\ & ⟹H_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right\}d{l}^p\le 0,r\in U,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & \left[\underset{r\in U}{\forall }{H}_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)\le H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\right.\hfill \\ \hfill & \left.\wedge \underset{r\in U}{\exists }{H}_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)<H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\right]\hfill \\ \hfill & ⟹H_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right\}d{l}^p<0,\mathit{for}\mathit{at}\mathit{least}\mathit{one}r\in U,\hfill \end{array}$$

(12)

and

$$\begin{array}{cc}\hfill & -H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\le 0\hfill \\ \hfill & ⟹H_{\delta }^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{\tau }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{{\tau }_{\alpha }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{ϵ}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{{ϵ}_{\beta }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right\}d{l}^p\le 0,q\in V,\hfill \end{array}$$

(13)

hold, for all $(\tau ,ϵ)\in A\times Y$, then $({\lambda }_p,\delta )$ is named strong-pseudo-quasi-H-type I at $(\overline{\tau },\overline{ϵ})\in A\times Y$ on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

If (11), (12) and (13) are fulfilled for each $(\overline{\tau },\overline{ϵ})\in A\times Y$, then the pair $({\lambda }_p,\delta )$ is named strong-pseudo-quasi-H-type I on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

Definition 11. 

For $(\overline{\tau },\overline{ϵ})\in A\times Y$, if there exists a smooth vector-valued functional $H_{{\lambda }_p}=\left(H_{{\lambda }_p^1},\dots ,H_{{\lambda }_p^v}\right):\mathbb{R}\to {\mathbb{R}}^v$ with $H_{{\lambda }_p^r}:Im\left(F^r\right)\to \mathbb{R}$ as a strictly increasing function, a smooth vector-valued functional $H_{\delta }=\left(H_{{\delta }^1},\dots ,H_{{\delta }^u}\right):\mathbb{R}\to {\mathbb{R}}^u$ with $H_{{\delta }^q}:Im\left(G^q\right)\to \mathbb{R}$ as a strictly increasing function, $\nu :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^w$ with $\nu (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\nu |}_{l=a,b}=0$, and $\vartheta :D\times {\mathbb{R}}^{2n}\times {\mathbb{R}}^{2m}\to {\mathbb{R}}^x$ with $\vartheta (l,\overline{\tau }\left(l\right),\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right),\overline{ϵ}\left(l\right)){=\vartheta |}_{l=a,b}=0$, such that

$$\begin{array}{cc}& \left[\underset{r\in U}{\forall }{H}_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)\le H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\right.\hfill \\ & \left.\wedge \underset{r\in U}{\exists }{H}_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)<H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ⟹H_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right\}d{l}^p<0,r\in U,\hfill \end{array}$$

(14)

and

$$\begin{array}{cc}\hfill & -H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\le 0\hfill \\ \hfill & ⟹H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{\tau }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{{\tau }_{\alpha }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\delta }_{ϵ}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{{ϵ}_{\beta }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right\}d{l}^p\le 0,q\in V,\hfill \end{array}$$

(15)

hold, for all $(\tau ,ϵ)\in A\times Y$, then $({\lambda }_p,\delta )$ is named weak-strictly-pseudo-quasi-H-type I at $(\overline{\tau },\overline{ϵ})\in A\times Y$ on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

If (14) and (15) are fulfilled for each $(\overline{\tau },\overline{ϵ})\in A\times Y$, then the pair $({\lambda }_p,\delta )$ is named weak-strictly-pseudo-quasi-H-type I on $A\times Y$ related to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

3. Main Results

In this section, we associate with (Primal) a dual problem, named (Dual), in order to study and investigate the relationships between the set of solutions for these two variational control problems. In short, duality provides a connection between two unconstrained or constrained extremization models, namely, the primal problem (a minimization/maximization problem), and the dual problem (a maximization/minimization model). In fact, the main goal is to ensure that the existence of a solution to one problem ensures the existence of a solution to onother problem (under various assumptions).

Thus, we introduce the following:

$$\begin{array}{cc}& \left(\mathrm{Dual}\right)\underset{(o,z)}{max}{\int }_C{\lambda }_p\left({\gamma }_{oz}\left(l\right)\right)d{l}^p=\left({\int }_C{\lambda }_p^1\left({\gamma }_{oz}\left(l\right)\right)d{l}^p,\dots ,{\int }_C{\lambda }_p^v\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\hfill \\ & \mathrm{subject}\mathrm{to}\hfill \\ & \sum _{r=1}^v{\theta }_r{H}_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\left[{\left({\lambda }_p\right)}_o^r\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p\right)}_{o_{\alpha }}^r\left({\gamma }_{oz}\left(l\right)\right)\right]\hfill \\ & +\sum _{q=1}^u{\sigma }_q\left(l\right)H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\left[{\delta }_o^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{o_{\alpha }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]=0,l\in D,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \sum _{r=1}^v{\theta }_r{H}_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\left[{\left({\lambda }_p\right)}_z^r\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p\right)}_{z_{\beta }}^r\left({\gamma }_{oz}\left(l\right)\right)\right]\hfill \\ \hfill & +\sum _{q=1}^u{\sigma }_q\left(l\right)H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\left[{\delta }_z^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{z_{\beta }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]=0,l\in D,\hfill \\ \hfill & {\sigma }_q\left(l\right)H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\ge 0,l\in D,\hfill \\ \hfill & \theta \in R^v,\theta \ge 0,\sigma \left(l\right)\in R^u,\sigma \left(l\right)\geqq 0,o\left(a\right)=\varphi ,o\left(b\right)=\psi ,\hfill \end{array}$$

where ${\lambda }_p$ and $\delta$ are defined as in the previous section.

Let B be the feasible solution set in (Dual),

$$B=\left\{\right(o,z,\theta ,\sigma ):(o,z)\in A\times Y,$$

$$\mathrm{verifying}\mathrm{the}\mathrm{constraints}\mathrm{of}\left(\mathrm{Dual}\right)\mathrm{for}\mathrm{all}l\in D\}.$$

The next result presents a weak-type duality between the variational models considered. Specifically, under various assumptions of the pair $({\lambda }_p,\delta )$, this theorem establishes that the value of the cost functional associated with the primal model cannot be greater than the value of the cost functional associated with the dual model.

Theorem 1. 

Let $(\tau ,ϵ)\in \mathcal{K}$ and $(o,z,\theta ,\sigma )\in B$ and assume that one of the following hypotheses is satisfied:

(a) the pair $({\lambda }_p,\delta )$ is strictly H-type I at $(o,z)$ with respect to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ;

(b) the pair $({\lambda }_p,\delta )$ is strictly-pseudo-quasi-H-type I at $(o,z)$ with respect to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ;

(c) the pair $({\lambda }_p,\delta )$ is strong-pseudo-quasi-H-type I at $(o,z)$ with respect to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

Then, the following relations cannot hold

$${\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\le {\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p,\mathit{for}\mathit{each}r\in U$$

(16)

and

$${\int }_C{\lambda }_p^{r^{*}}\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p<{\int }_C{\lambda }_p^{r^{*}}\left({\gamma }_{oz}\left(l\right)\right)d{l}^p,\mathit{for}\mathit{some}{r}^{*}\in U.$$

(17)

Proof. 

Let $(\tau ,ϵ)$ and $(o,z,\theta ,\sigma )$ be feasible solutions in the considered multiple-objective models (Primal) and (Dual), respectively. We proceed by contradiction and suppose, contrary to the result, that (16) and (17) are satisfied. Further, we consider that we are in hypothesis a). Since the pair $({\lambda }_p,\delta )$ is strictly H-type I at $(o,z)$ with respect to $H_{{\lambda }_p},H_{\delta },\nu$ and $\vartheta$, the following inequalities are satisfied

$$\begin{array}{cc}\hfill & H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)-H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & >H_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p\right)}_o^r\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p\right)}_{o_{\alpha }}^r\left({\gamma }_{oz}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p\right)}_z^r\left({\gamma }_{oz}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p\right)}_{z_{\beta }}^r\left({\gamma }_{oz}\left(l\right)\right)\right\}d{l}^p,r\in U,\hfill \end{array}$$

(18)

and

$$\begin{array}{cc}\hfill & -H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & \ge H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\delta }_o^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{o_{\alpha }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\delta }_z^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{z_{\beta }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]\right\}d{l}^p,q\in V.\hfill \end{array}$$

(19)

Since every $H_{{\lambda }_p^r},r\in U$ is a strictly increasing functional, the inequalities (16) and (17) yield

$$H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)\le H_{{\lambda }_p^r}\left({\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right),r\in U,$$

(20)

and

$$H_{{\lambda }_p^{r^{*}}}\left({\int }_C{\lambda }_p^{r^{*}}\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p\right)<H_{{\lambda }_p^{r^{*}}}\left({\int }_C{\lambda }_p^{r^{*}}\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right),\mathrm{for}\mathrm{some}{r}^{*}\in U.$$

(21)

By (18), (20) and (21), it follows that

$$\begin{array}{cc}\hfill & H_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right){\int }_C\left[{\left[\nu \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p\right)}_o^r\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p\right)}_{o_{\alpha }}^r\left({\gamma }_{oz}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p\right)}_z^r\left({\gamma }_{oz}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p\right)}_{z_{\beta }}^r\left({\gamma }_{oz}\left(l\right)\right)\right\}d{l}^p<0,r\in U.\hfill \end{array}$$

(22)

Multiplying each inequality (22) by ${\theta }_r,r\in U$, and then adding both sides of the obtained inequalities, we obtain

$$\begin{array}{cc}\hfill & \sum _{r=1}^v{\theta }_r{H}_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p\right)}_o^r\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p\right)}_{o_{\alpha }}^r\left({\gamma }_{oz}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\left({\lambda }_p\right)}_z^r\left({\gamma }_{oz}\left(l\right)\right)\right]-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p\right)}_{z_{\beta }}^r\left({\gamma }_{oz}\left(l\right)\right)\right\}d{l}^p<0.\hfill \end{array}$$

(23)

Multiplying each inequality

$$\begin{array}{cc}\hfill & -H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & \ge H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\delta }_o^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{o_{\alpha }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\delta }_z^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{z_{\beta }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]\right\}d{l}^p,q\in V,\hfill \end{array}$$

by ${\sigma }_q\left(l\right)\ge 0,q\in V$, and then adding both sides of the obtained inequalities, it follows

$$\begin{array}{cc}\hfill & -\sum _{q=1}^u{\sigma }_q\left(l\right)H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & \ge \sum _{q=1}^u{\sigma }_q\left(l\right)H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\delta }_o^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{o_{\alpha }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\delta }_z^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{z_{\beta }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]\right\}d{l}^p.\hfill \end{array}$$

(24)

Using the feasibility of $(o,z,\theta ,\sigma )$ in (Dual) together with (24), we obtain

$$\begin{array}{cc}\hfill & \sum _{q=1}^u{\sigma }_q\left(l\right)H_{\delta q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\delta }_o^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{o_{\alpha }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]\right.\hfill \\ \hfill & \left.{\left[\vartheta \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left[{\delta }_z^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{z_{\beta }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]\right\}d{l}^p\le 0.\hfill \end{array}$$

(25)

Adding both sides of (23) and (25), it results that the inequality

$$\begin{array}{cc}& {\int }_C{\left[\nu \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left\{\sum _{r=1}^v{\theta }_r{H}_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\left[{\left({\lambda }_p\right)}_o^r\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p\right)}_{o_{\alpha }}^r\left({\gamma }_{oz}\left(l\right)\right)\right]\right.\hfill \\ & \left.+\sum _{q=1}^u{\sigma }_q\left(l\right)H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\left[{\delta }_o^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{o_{\alpha }}^q\left({\gamma }_{oz}\left(l\right)\right)\right]\right\}d{l}^p\hfill \\ & +{\int }_C{\left[\vartheta \left({\gamma }_{\tau ϵoz}\left(l\right)\right)\right]}^T\left\{\sum _{r=1}^v{\theta }_r{H}_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\left[{\left({\lambda }_p\right)}_z^r\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p\right)}_{z_{\beta }}^r\left({\gamma }_{oz}\left(l\right)\right)\right]\right.\hfill \\ & \left.+\sum _{q=1}^u{\sigma }_q\left(l\right)H_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{oz}\left(l\right)\right)d{l}^p\right)\left[{\delta }_z^q\left({\gamma }_{oz}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_z^q\left({\gamma }_{oz}\left(l\right)\right)\right]\right\}d{l}^p<0\hfill \end{array}$$

is verified, contradicting the feasibility of $(o,z,\theta ,\sigma )$ in (Dual). Thus, we complete the proof under hypothesis (a). In a similar way, we can prove the current theorem under each of the hypotheses given in (b) and (c). □

If we consider weaker generalized invexity hypotheses, then a weaker result is true, as follows:

Theorem 2. 

Let $(\tau ,ϵ)\in \mathcal{K}$ and $(o,z,\theta ,\sigma )\in B$ and assume that one of the following hypotheses is satisfied:

(a) the pair $({\lambda }_p,\delta )$ is H-type I at $(o,z)$ with respect to $H_{{\lambda }_p}$, $H_{\delta },\nu$ and ϑ;

(b) the pair $({\lambda }_p,\delta )$ is pseudo-quasi-H-type I at $(o,z)$ with respect to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ;

(c) the pair $({\lambda }_p,\delta )$ is weak-strictly-pseudo-quasi-H-type I at $(o,z)$ with respect to $H_{{\lambda }_p},H_{\delta },\nu$ and ϑ.

Then the following relation cannot hold

$${\int }_C{\lambda }_p^r\left({\gamma }_{\tau ϵ}\left(l\right)\right)d{l}^p<{\int }_C{\lambda }_p^r\left({\gamma }_{oz}\left(l\right)\right)d{l}^p,\mathit{for}\mathit{each}r\in U.$$

The following two results present a strong-type duality between the variational models considered. Concretely, under the same hypotheses formulated in the above two theorems and considering $(\overline{\tau },\overline{ϵ})$ to be an efficient (weakly efficient) point of (Primal), we state that $(\overline{\tau },\overline{ϵ},\overline{\theta },\overline{\sigma })$ is an (weakly efficient) efficient point of (Dual). Since the proofs for these two theorems are (in large part) the same, we provide only the proof of Theorem 4.

Theorem 3. 

Let $(\overline{\tau },\overline{ϵ})$ be an (weakly efficient) efficient point of (Primal), that is, the following necessary efficiency conditions

$$\begin{array}{cc}\hfill & \sum _{r=1}^v{\overline{\theta }}_r{H}_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\hfill \\ \hfill & +\sum _{q=1}^u{\overline{\sigma }}_q{H}_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\left[{\delta }_{\tau }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{{\tau }_{\alpha }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]=0,l\in D,\hfill \\ \hfill & \sum _{r=1}^v{\overline{\theta }}_r{H}_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\hfill \\ \hfill & +\sum _{q=1}^u{\overline{\sigma }}_q{H}_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\left[{\delta }_{ϵ}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{{ϵ}_{\beta }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]=0,l\in D,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\overline{\sigma }}_q\left(l\right)H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)=0,l\in D,q\in V,\hfill \\ \hfill & \overline{\theta }\ge 0,{\overline{\theta }}^{T}e=1,\overline{\sigma }\left(l\right)\geqq 0,\hfill \end{array}$$

are satisfied at this point. Then $(\overline{\tau },\overline{ϵ},\overline{\theta },\overline{\sigma })$ is a feasible solution of (Dual). In addition, if Theorem 1 holds, then $(\overline{\tau },\overline{ϵ},\overline{\theta },\overline{\sigma })$ is an (weakly efficient) efficient point of (Dual).

Theorem 4. 

Let $(\overline{\tau },\overline{ϵ})$ be a properly efficient point of (Primal), that is, the following necessary efficiency conditions

$$\begin{array}{cc}\hfill & \sum _{r=1}^v{\overline{\theta }}_r{H}_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\left[{\left({\lambda }_p^r\right)}_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\left({\lambda }_p^r\right)}_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\hfill \\ \hfill & +\sum _{q=1}^u{\overline{\sigma }}_q{H}_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\left[{\delta }_{\tau }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\alpha }}}{\delta }_{{\tau }_{\alpha }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]=0,l\in D,\hfill \\ \hfill & \sum _{r=1}^v{\overline{\theta }}_r{H}_{{\lambda }_p^r}^{\prime }\left({\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\left[{\left({\lambda }_p^r\right)}_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\left({\lambda }_p^r\right)}_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\hfill \\ \hfill & +\sum _{q=1}^u{\overline{\sigma }}_q{H}_{{\delta }^q}^{\prime }\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)\left[{\delta }_{ϵ}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-{\displaystyle \frac{\partial }{\partial l^{\beta }}}{\delta }_{{ϵ}_{\beta }}^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]=0,l\in D,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\overline{\sigma }}_q\left(l\right)H_{{\delta }^q}\left({\int }_C{\delta }^q\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)d{l}^p\right)=0,l\in D,q\in V,\hfill \\ \hfill & \overline{\theta }\ge 0,{\overline{\theta }}^{T}e=1,\overline{\sigma }\left(l\right)\geqq 0,\hfill \end{array}$$

are satisfied at this point, and assume the hypotheses in Theorem 1 are fulfilled. Then $(\overline{\tau },\overline{ϵ},\overline{\theta },\overline{\sigma })$ is a feasible solution of (Dual). In addition, $(\overline{\tau },\overline{ϵ},\overline{\theta },\overline{\sigma })$ is a properly efficient point of (Dual) and the objective values at these points are equal.

Proof. 

Since $(\overline{\tau },\overline{ϵ})$ is a properly efficient point of (Primal), there exist $\overline{\theta }\in R^v,\overline{\theta }\ge 0$ and a piecewise smooth function $\overline{\sigma }(·):D\to R^u$ such that the above-mentioned conditions are satisfied at this point. Thus, $(\overline{\tau },\overline{ϵ},\overline{\theta },\overline{\sigma })$ is feasible in the dual model (Dual). By Theorem 1, it follows that $(\overline{\tau },\overline{ϵ},\overline{\theta },\overline{\sigma })$ is an efficient point of (Dual). Next, we prove that $(\overline{\tau },\overline{ϵ},\overline{\theta },\overline{\sigma })$ is a properly efficient point of (Dual) by the technique of contradiction. Then, there exists $(\tilde{o},\tilde{z},\tilde{\theta },\tilde{\sigma })\in B$ and $r^{*}\in U$ such that the following inequality

$${\int }_C{\lambda }_p^{r^{*}}\left({\gamma }_{\tilde{o}\tilde{z}}\left(l\right)\right)d{l}^p-{\int }_C{\lambda }_p^{r^{*}}\left({\gamma }_{\overline{\tau ϵ}}\left(l\right)\right)d{l}^p$$

$$>L\left({\int }_C{\lambda }_p^k\left({\gamma }_{\overline{\tau ϵ}}\left(l\right)\right)d{l}^p-{\int }_C{\lambda }_p^k\left({\gamma }_{\tilde{o}\tilde{z}}\left(l\right)\right)d{l}^p\right)$$

(26)

is valid for $L>0$ and k, verifying

$${\int }_C{\lambda }_p^k\left({\gamma }_{\overline{\tau ϵ}}\left(l\right)\right)d{l}^p>{\int }_C{\lambda }_p^k\left({\gamma }_{\tilde{o}\tilde{z}}\left(l\right)\right)d{l}^p.$$

(27)

We consider the index set $U=U_1\cup U_2$ and denote by $U_1$ the set of indices of cost functionals satisfying the inequality (27). By $U_2$ we denote the set of indices of cost functionals defined as follows $U_2=U\setminus \left(U_1\cup \left\{r^{*}\right\}\right)$. The inequality (26) is satisfied for all $L>0$. Then, we set $L>{\displaystyle \frac{{\overline{\theta }}_k}{{\overline{\theta }}_{r^{*}}}}\left|U_1\right|$, where $\left|U_1\right|$ denotes the cardinal of the set $U_1$. In consequence, (26) and (27) yield

$$\begin{array}{cc}\hfill & {\overline{\theta }}_{r^{*}}\left({\int }_C{\lambda }_p^{r^{*}}\left({\gamma }_{\overline{\tau ϵ}}\left(l\right)\right)d{l}^p-{\int }_C{\lambda }_p^{r^{*}}\left({\gamma }_{\tilde{o}\tilde{z}}\left(l\right)\right)d{l}^p\right)\hfill \\ \hfill & >\sum _{k\in U_1}{\overline{\theta }}_k\left({\int }_C{\lambda }_p^k\left({\gamma }_{\overline{\tau ϵ}}\left(l\right)\right)d{l}^p-{\int }_C{\lambda }_p^k\left({\gamma }_{\tilde{o}\tilde{z}}\left(l\right)\right)d{l}^p\right).\hfill \end{array}$$

(28)

By the definition of the set $U_2$, (26), (27) and (28), it follows that

$$\begin{array}{cc}& \sum _{r=1}^v{\overline{\theta }}_r{\int }_C{\lambda }_p^r\left({\gamma }_{\overline{\tau ϵ}}\left(l\right)\right)d{l}^p\hfill \\ & ={\overline{\theta }}_{r^{*}}{\int }_C{\lambda }_p^{r^{*}}\left({\gamma }_{\overline{\tau ϵ}}\left(l\right)\right)d{l}^p+\sum _{k\in U_1}{\overline{\theta }}_k{\int }_C{\lambda }_p^k\left({\gamma }_{\overline{\tau ϵ}}\left(l\right)\right)d{l}^p+\sum _{k\in U_2}{\overline{\theta }}_k{\int }_C{\lambda }_p^k\left({\gamma }_{\overline{\tau ϵ}}\left(l\right)\right)d{l}^p\hfill \\ & <{\overline{\theta }}_{r^{*}}{\int }_C{\lambda }_p^{r^{*}}\left({\gamma }_{\tilde{o}\tilde{z}}\left(l\right)\right)d{l}^p+\sum _{k\in U_1}{\overline{\theta }}_k{\int }_C{\lambda }_p^k\left({\gamma }_{\tilde{o}\tilde{v}}\left(l\right)\right)d{l}^p+\sum _{k\in U_2}{\overline{\theta }}_k{\int }_C{\lambda }_p^k\left({\gamma }_{\tilde{o}\tilde{v}}\left(l\right)\right)d{l}^p\hfill \\ & =\sum _{r=1}^v{\overline{\theta }}_r{\int }_C{\lambda }_p^r\left({\gamma }_{\tilde{o}\tilde{z}}\left(l\right)\right)d{l}^p.\hfill \end{array}$$

This is a contradiction with Theorem 1. Hence, $(\overline{\tau },\overline{ϵ},\overline{\theta },\overline{\sigma })$ is a properly efficient point of (Dual), and the optimal cost functional values are equal in the dual and the primal problems. □

In the following, we illustrate the main findings of the current paper with some numerical examples.

Illustrative example 1. Let $v=j=2,w=x=1$, $A=\{\tau :D_{a,b}\to [-2,2]\subset \mathbb{R}\},Y=\{ϵ:D_{a,b}\to [-1,1]\subset \mathbb{R}\}$, where $D_{a,b}$ is a square fixed with the diagonally opposite points $a=\left(a^1,a^2\right)=(-2,-2)$ and $b=\left(b^1,b^2\right)=(2,2)$ in ${\mathbb{R}}^2$. Now, we consider the vector-valued functional $\lambda :D_{a,b}\times A\times Y\mapsto \mathbb{R}$, defined by $\lambda (l,\tau (l),ϵ(l\left)\right)=ln\left[\tau \right(l)+ϵ(l)+5]$, that generates the curvilinear integral functional

$${\displaystyle F:A\times Y\to \mathbb{R},}$$

$${\displaystyle F(\tau ,ϵ)={\int }_C\lambda (l,\tau \left(l\right),ϵ\left(l\right))(d{l}^1+d{l}^2)={\int }_{C}ln[\tau \left(l\right)+ϵ\left(l\right)+5](d{l}^1+d{l}^2).}$$

Also, let us define

$$\overline{\tau }\left(l\right)={\displaystyle \frac{l^1+l^2}{2}},\overline{ϵ}\left(l\right)={\displaystyle \frac{2l^1+l^2}{6}},l=(l^1,l^2)\in D_{a,b}$$

and consider $l^1=l^2=1$. We have

$$\begin{array}{cc}& {\int }_C\lambda (l,\tau \left(l\right),ϵ\left(l\right))(d{l}^1+d{l}^2)-{\int }_C\lambda (l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))(d{l}^1+d{l}^2)\hfill \\ & -{\int }_C[{\displaystyle \frac{\partial \lambda }{\partial \tau \left(l\right)}}(l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))(\tau \left(l\right)-\overline{\tau }\left(l\right))\hfill \\ & +{\displaystyle \frac{\partial \lambda }{\partial ϵ\left(l\right)}}(l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))(ϵ\left(l\right)-\overline{ϵ}\left(l\right))](d{l}^1+d{l}^2)\hfill \\ & ={\int }_C\left(ln[\tau \left(l\right)+ϵ\left(l\right)+5]-ln{\displaystyle \frac{13}{2}}-{\displaystyle \frac{2}{13}}\left[\tau \left(l\right)+ϵ\left(l\right)-{\displaystyle \frac{3}{2}}\right]\right)(d{l}^1+d{l}^2)\hfill \\ & \ngeqq 0,\forall (\tau ,ϵ)\in A\times Y.\hfill \end{array}$$

Thus, the above-mentioned real-valued double integral functional is not convex at $\left(1,{\displaystyle \frac{1}{2}}\right)\in A\times Y$. On the other hand, if we consider the strictly increasing function $H_{\lambda }:Im\left(F\right)\mapsto \mathbb{R}$ defined by $H_{\lambda }\left(\lambda (l,\tau \left(l\right),ϵ\left(l\right))\right)=e^{2\lambda (l,\tau (l),ϵ(l\left)\right)}$, then we obtain

$$\begin{array}{cc}& H_{\lambda }\left({\int }_C\lambda \left({\gamma }_{\tau ϵ}\left(l\right)\right)(d{l}^1+d{l}^2)\right)-H_{\lambda }\left({\int }_C\lambda \left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)(d{l}^1+d{l}^2)\right)\hfill \\ & -H_{\lambda }^{\prime }\left({\int }_C\lambda \left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)(d{l}^1+d{l}^2)\right){\int }_C\left\{{\left[\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\tau }_{\tau }\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)-D_{\alpha }{\tau }_{{\tau }_{\alpha }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]\right.\hfill \\ & \left.+{\left[\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)\right]}^T\left[{\tau }_{ϵ}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right]-D_{\beta }{\tau }_{{ϵ}_{\beta }}\left({\gamma }_{\overline{\tau }\overline{ϵ}}\left(l\right)\right)\right\}(d{l}^1+d{l}^2)\hfill \end{array}$$

$$\begin{array}{cc}& ={\int }_C\left({\left(\tau \left(l\right)+ϵ\left(l\right)+5\right)}^2-{\left({\displaystyle \frac{13}{2}}\right)}^2-13\left[\tau \left(l\right)+ϵ\left(l\right)-{\displaystyle \frac{3}{2}}\right]\right)(d{l}^1+d{l}^2)\hfill \\ & \ge 0,\forall (\tau ,ϵ)\in A\times Y,\hfill \end{array}$$

with $\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)=\tau \left(l\right)-\overline{\tau }\left(l\right)$ and $\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)=ϵ\left(l\right)-\overline{ϵ}\left(l\right)$. Similar computations can be written for

$${\displaystyle G:A\times Y\to \mathbb{R},}$$

$${\displaystyle G(\tau ,ϵ)={\int }_C\delta (l,\tau \left(l\right),ϵ\left(l\right))(d{l}^1+d{l}^2)={\int }_{C}ln[1+3o\left(l\right)](d{l}^1+d{l}^2),}$$

with $H_{\delta }\left(b\right)=b$, $\nu \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)=\tau \left(l\right)-\overline{\tau }\left(l\right)$, $\vartheta \left({\gamma }_{\tau ϵ\overline{\tau ϵ}}\left(l\right)\right)=ϵ\left(l\right)-\overline{ϵ}\left(l\right)$. Thus, the curvilinear integral functionals given above form the pair $(\lambda ,\delta )$ and it is H-type I at $\left(1,{\displaystyle \frac{1}{2}}\right)\in A\times Y$. Moreover, since all the hypotheses in Theorem 4 are satisfied, the Lagrange multipliers $\overline{\theta }={\displaystyle \frac{1}{2}},\overline{\sigma }={\displaystyle \frac{2}{3}}$, then $(\overline{\tau },\overline{ϵ},\overline{\theta },\overline{\sigma })$ is a properly efficient point of the associated (Dual) and the objective values at these points are equal.

Illustrative example 2. Let $\tau :D=D_{l_0,l_1}\to \mathbb{R}$ and $ϵ:D=D_{l_0,l_1}\to \mathbb{R}$ and ${\mathsf{\Omega }}_{0,1}$ be a piecewise curve linking the diagonally opposite points $l_0=(l_0^1,l_0^2)=(0,0)$ and $l_1=(l_1^1,l_1^2)=(1,1)$ în ${\mathbb{R}}^2$. Let us define the multidimensional first-order PDE-constrained control problem given as follows:

$$\left(\mathrm{MCP}2\right)\underset{(\tau ,ϵ)}{min}\left({\int }_{{\mathsf{\Omega }}_{0,1}}(1+3ϵ\left(l\right))(d{l}^1+d{l}^2),{\int }_{{\mathsf{\Omega }}_{0,1}}(3e^{{\displaystyle \frac{1}{4}}}+4-2ϵ\left(l\right))(d{l}^1+d{l}^2)\right)$$

subject to

$${\displaystyle \frac{\partial \tau \left(l\right)}{\partial l^1}}={log}_e(ϵ\left(l\right)-2),{\displaystyle \frac{\partial \tau \left(l\right)}{\partial l^2}}={log}_e(ϵ\left(l\right)-2),$$

(29)

$${log}_e\left(\tau \left(l\right)-{\displaystyle \frac{5}{2}}\right)\le 0,$$

(30)

$$\tau (0,0)=3,\tau (1,1)={\displaystyle \frac{7}{2}}.$$

(31)

We aim to find the control function $ϵ\left(l\right)$ (that determines the state function $\tau \left(l\right)$) which satisfies the dynamical system (29) with respect to boundary condition (31), such as to minimize the objective vector functional. Also, we are interested in affine state functions.

Consider $(\overline{\tau },\overline{ϵ})$ is an efficient solution to (MCP2). Then, by the classical necessary efficiency conditions, there exist piecewise smooth functions $\mu =\mu \left(l\right)\in {\mathbb{R}}_{+}$ and ${\mathsf{\Lambda }}^{\alpha }={\mathsf{\Lambda }}^{\alpha }\left(l\right)\in \mathbb{R},\alpha =1,2$, satisfying

$$\mu \left[{\displaystyle \frac{1}{\tau \left(l\right)-\frac{5}{2}}}\right]+{\displaystyle \frac{\partial {\mathsf{\Lambda }}^1}{\partial l^1}}+{\displaystyle \frac{\partial {\mathsf{\Lambda }}^2}{\partial l^2}}=0,$$

(32)

$${\displaystyle \frac{3}{1+3ϵ\left(l\right)}}+{\displaystyle \frac{-2}{3e^{\frac{1}{4}}+4-2ϵ\left(l\right)}}+{\mathsf{\Lambda }}^1\left[{\displaystyle \frac{1}{ϵ\left(l\right)-2}}\right]+{\mathsf{\Lambda }}^2\left[{\displaystyle \frac{1}{ϵ\left(l\right)-2}}\right]=0,$$

(33)

$$\mu \left[{log}_e\left(\tau \left(l\right)-{\displaystyle \frac{5}{2}}\right)\right]=0,$$

(34)

$$\mu \ge 0.\left(35\right)$$

(35)

Now, we introduce the following increasing functions which help to simplify the previous problem: $H_{\lambda }\left(a\right)=e^a{H}_{\delta }\left(a\right)=e^{2a},a\in \mathbb{R}$.

From the efficiency of $(\overline{\tau },\overline{ϵ})$ to (MCP2), there exist the piecewise smooth functions $\mu =\mu \left(l\right)\in {\mathbb{R}}_{+}$ and ${\mathsf{\Lambda }}^{\alpha }={\mathsf{\Lambda }}^{\alpha }\left(l\right)\in \mathbb{R},\alpha =1,2$, such that the H-necessary efficiency conditions are satisfied at $(\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))$ as follows:

$$\mu +{\displaystyle \frac{\partial {\mathsf{\Lambda }}^1}{\partial l^1}}+{\displaystyle \frac{\partial {\mathsf{\Lambda }}^2}{\partial l^2}}=0,$$

(36)

$$1+2{\mathsf{\Lambda }}^1(\overline{ϵ}\left(l\right)-2)+2{\mathsf{\Lambda }}^2(\overline{ϵ}\left(l\right)-2)=0,$$

(37)

$${\mu }^{\beta }[\tau \left(l\right)-\overline{\tau }\left(l\right)]\le 0,$$

(38)

$$\mu \ge 0,\left(39\right)$$

(39)

for all $l\in D_{l_0,l_1}$, except at discontinuities. Indeed, the above system is easier to study in comparison to solving the system (32)–(35), which is an advantage of H-necessary efficiency conditions. Now, taking into account relations (38) and (39), we consider $\mu =0$.

Case 1. Assume that $\mathsf{\Lambda }=\left({\mathsf{\Lambda }}^1,{\mathsf{\Lambda }}^2\right)$ are constant functions, say ${\mathsf{\Lambda }}^1={\mathbf{a}}_1$ and ${\mathsf{\Lambda }}^2={\mathbf{a}}_2$, where ${\mathbf{a}}_1,{\mathbf{a}}_2\in R$. Therefore, Equation (37) gives

$$\overline{ϵ}\left(l\right)={\displaystyle \frac{-1}{2({\mathbf{a}}_1+{\mathbf{a}}_2)}}+2,\forall l\in D_{l_0,l_1},$$

which shows that $\overline{ϵ}\left(l\right)$ is a constant function on $D_{l_0,l_1}$. Taking into account (29), it follows that

$$\overline{\tau }\left(l\right)={log}_e\left[{\displaystyle \frac{-1}{2({\mathbf{a}}_1+{\mathbf{a}}_2)}}\right]\left(l^1+l^2\right)+{\mathbf{a}}_3,{\mathbf{a}}_3\in {\mathbb{R}}_{+}.$$

Applying the boundary condition (31), we obtain ${\mathbf{a}}_3=3$ and ${\mathbf{a}}_1+{\mathbf{a}}_2=-{\displaystyle \frac{1}{2}}{e}^{-{\displaystyle \frac{1}{4}}}$. Therefore, we obtain

$$\overline{\tau }\left(l\right)={\displaystyle \frac{1}{4}}\left(l^1+l^2\right)+3,\overline{ϵ}\left(l\right)=e^{{\displaystyle \frac{1}{4}}}+2,\forall l\in D_{l_0,l_1}.$$

Now, one can clearly observe that the point $(\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))$ as given above is a $H-KT$ point associated with (MCP2) since there exist $\mathsf{\Lambda }=\left({\mathsf{\Lambda }}^1,{\mathsf{\Lambda }}^2\right)$ satisfying ${\mathsf{\Lambda }}^1+{\mathsf{\Lambda }}^2=-{\displaystyle \frac{1}{2}}{e}^{-{\displaystyle \frac{1}{4}}}$, and $\mu =0,\forall l\in D_{l_0,l_1}$, such that conditions (36)–(39) are fulfilled.

Further, let us show that $(\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))=\left({\displaystyle \frac{1}{4}}\left(l^1+l^2\right)+3,e^{{\displaystyle \frac{1}{4}}}+2\right)$ at $l^1=l^2=0$, is an efficient solution of (MCP2). For all $\left(\tau \right(l),ϵ(l\left)\right)\in \mathcal{D}$, we have the following:

$${\int }_{{\mathsf{\Omega }}_{0,1}}\{{\lambda }^1(l,\tau \left(l\right),ϵ\left(l\right))-{\lambda }^1(l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))\}(d{l}^1+d{l}^2)$$

$$={\int }_{{\mathsf{\Omega }}_{0,1}}\{{log}_e\left(1+3ϵ\left(l\right)\right)-{log}_e(1+3e^{{\displaystyle \frac{1}{4}}}+6)\}(d{l}^1+d{l}^2)$$

$$={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(1+3ϵ\left(l\right)\right)-{log}_e(7+3e^{{\displaystyle \frac{1}{4}}})\right\}(d{l}^1+d{l}^2),$$

(40)

and

$${\int }_{{\mathsf{\Omega }}_{0,1}}\{{\lambda }^2(l,\tau \left(l\right),ϵ\left(l\right))-{\lambda }^2(l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))\}(d{l}^1+d{l}^2)$$

$$={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}+4-2ϵ\left(l\right)\right)-{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}+4-2e^{{\displaystyle \frac{1}{4}}}-4\right)\right\}(d{l}^1+d{l}^2)$$

$$={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}+4-2ϵ\left(l\right)\right)-{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}-2e^{{\displaystyle \frac{1}{4}}}\right)\right\}(d{l}^1+d{l}^2)$$

$$={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}+4-2ϵ\left(l\right)\right)-{log}_e\left(e^{{\displaystyle \frac{1}{4}}}\right)\right\}(d{l}^1+d{l}^2)$$

$$={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}+4-2ϵ\left(l\right)\right)-{\displaystyle \frac{1}{4}}\right\}(d{l}^1+d{l}^2).$$

(41)

By solving Equation (29), we obtain

$$ϵ\left(l\right)=e^{\left({\displaystyle \frac{\partial \tau }{\partial l^1}}\right)}+2,\mathrm{or}ϵ\left(l\right)=e^{\left({\displaystyle \frac{\partial \tau }{\partial l^2}}\right)}+2,$$

and putting these values in Equations (40) and (41), we obtain

$${\int }_{{\mathsf{\Omega }}_{0,1}}\{{\lambda }^1(l,\tau \left(l\right),ϵ\left(l\right))-{\lambda }^1(l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))\}(d{l}^1+d{l}^2)$$

$$={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(1+3e^{\left({\displaystyle \frac{\partial \tau }{\partial l^1}}\right)}+6\right)-{log}_e\left(7+3e^{{\displaystyle \frac{1}{4}}}\right)\right\}(d{l}^1+d{l}^2)$$

$$={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(7+3e^{\left({\displaystyle \frac{\partial \tau }{\partial l^1}}\right)}\right)-{log}_e\left(7+3e^{{\displaystyle \frac{1}{4}}}\right)\right\}(d{l}^1+d{l}^2)$$

$$(\mathrm{or},{\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(7+3e^{\left({\displaystyle \frac{\partial \tau }{\partial l^2}}\right)}\right)-{log}_e\left(7+3e^{{\displaystyle \frac{1}{4}}}\right)\right\}(d{l}^1+d{l}^2),$$

(42)

and

$${\int }_{{\mathsf{\Omega }}_{0,1}}\{{\lambda }^2(l,\tau \left(l\right),ϵ\left(l\right))-{\lambda }^2(l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))\}(d{l}^1+d{l}^2)$$

$$={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}-2e^{\left({\displaystyle \frac{\partial \tau }{\partial l^1}}\right)}\right)-{\displaystyle \frac{1}{4}}\right\}(d{l}^1+d{l}^2)$$

$$(\mathrm{or},\left.{\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}-2e^{\left({\displaystyle \frac{\partial \tau }{\partial l^2}}\right)}\right)-{\displaystyle \frac{1}{4}}\right\}(d{l}^1+d{l}^2)\right).$$

(43)

Now, we have to find the state function $\tau \left(l\right)$ which satisfies the boundary conditions (34). From the equality constraints (29), we have

$${\displaystyle \frac{\partial \tau }{\partial l^1}}\left(l\right)={\displaystyle \frac{\partial \tau }{\partial l^2}}\left(l\right),\forall l=\left(l^1,l^2\right)\in D_{l_0,l_1}·$$

Applying the necessary Euler–Lagrange PDE, we obtain

$$\tau \left(l\right)=\mathsf{\Phi }\left(l^1+l^2\right)+\psi ,\mathsf{\Phi },\psi \in R,\forall l\in D_{l_0,l_1}.$$

Applying boundary condition (31) on the above solution, we obtain that $\mathsf{\Phi }={\displaystyle \frac{1}{4}}$ and $\psi =3$. Hence,

$$\tau \left(l\right)={\displaystyle \frac{1}{4}}\left(l^1+l^2\right)+3.$$

Therefore, the inequalities (42) and (43) together with the solution $\tau \left(l\right)={\displaystyle \frac{1}{4}}\left(l^1+l^2\right)+3$ conclude that

$$\begin{array}{cc}& {\int }_{{\mathsf{\Omega }}_{0,1}}\{{\lambda }^1(l,\tau \left(l\right),ϵ\left(l\right))-{\lambda }^1(l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(7+3e^{\left({\displaystyle \frac{\partial \tau }{\partial l^1}}\right)}\right)-{log}_e\left(7+3e^{{\displaystyle \frac{1}{4}}}\right)\right\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}\{{log}_e\left(7+3e^{{\displaystyle \frac{1}{4}}}\right)-{log}_e\left(7+3e^{{\displaystyle \frac{1}{4}}}\right\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}0(d{l}^1+d{l}^2)=0(\ge 0),\hfill \\ & (\mathrm{or},{\int }_{{\mathsf{\Omega }}_{0,1}}\{{\lambda }^1(l,\tau \left(l\right),ϵ\left(l\right))-{\lambda }^1(l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(7+3e^{\left({\displaystyle \frac{\partial \tau }{\partial l^2}}\right)}\right)-{log}_e\left(7+3e^{{\displaystyle \frac{1}{4}}}\right)\right\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}\{{log}_e\left(7+3e^{{\displaystyle \frac{1}{4}}}\right)-{log}_e\left(7+3e^{{\displaystyle \frac{1}{4}}}\right)\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}0(d{l}^1+d{l}^2)=0(\ge 0)).\hfill \end{array}$$

Also,

$$\begin{array}{cc}& {\int }_{{\mathsf{\Omega }}_{0,1}}\{{\lambda }^2(l,\tau \left(l\right),ϵ\left(l\right))-{\lambda }^2(l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}-2e^{\left({\displaystyle \frac{\partial \tau }{\partial l^1}}\right)}\right)-{\displaystyle \frac{1}{4}}\right\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}-2e^{{\displaystyle \frac{1}{4}}}\right)-{\displaystyle \frac{1}{4}}\right\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}0(d{l}^1+d{l}^2)=0(\ge 0),\hfill \\ & (\mathrm{or},{\int }_{{\mathsf{\Omega }}_{0,1}}\{{\lambda }^2(l,\tau \left(l\right),ϵ\left(l\right))-{\lambda }^2(l,\overline{\tau }\left(l\right),\overline{ϵ}\left(l\right))\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}-2e^{\left({\displaystyle \frac{\partial \tau }{\partial l^2}}\right)}\right)-{\displaystyle \frac{1}{4}}\right\}(d{l}^1+d{l}^2)\hfill \\ & ={\int }_{{\mathsf{\Omega }}_{0,1}}\left\{{log}_e\left(3e^{{\displaystyle \frac{1}{4}}}-2e^{{\displaystyle \frac{1}{4}}}\right)-{\displaystyle \frac{1}{4}}\right\}(d{l}^1+d{l}^2)\hfill \\ & \left.={\int }_{{\mathsf{\Omega }}_{0,1}}0(d{l}^1+d{l}^2)=0(\ge 0)\right).\hfill \end{array}$$

Case 2. If we assume that $\mathsf{\Lambda }=\left({\mathsf{\Lambda }}^1,{\mathsf{\Lambda }}^2\right)$ are not constant functions, we obtain a contradiction with the hypothesis (we are interested in affine state functions).

In conclusion, we have provided a detailed computation on a concrete numerical example, establishing the key steps in calculating the efficient solution.

4. Conclusions

In this study, we have introduced and studied a new class of multi-objective control models generated by functionals driven by path-independent curvilinear integrals, which imply partial derivatives of the control variable. Concretely, we have analyzed its set of solutions by considering a dual problem. Thus, various results of duality type have been formulated and proved in order to study the relationships between the set of solutions for the two considered variational control problems. Namely, in Theorem 1 and Theorem 2, under some H-type I hypotheses of the functionals used in our models, we have established that the value of the objective functional in the primal model cannot overcome the value of the objective functional in the dual problem. The next two theorems (see Theorem 3 and Theorem 4) provide a strong type duality for the primal and dual problems. As further research directions based on this paper, the authors suggest the well-posedness study and saddle-point-type efficiency criteria associated with this class of optimization models. Also, we can extend the results to distributed cases where the computations are distributed over a multi-agent network.